The country’s leading gay rights groups and donors, after a decade focused on legalizing same-sex marriage, are embarking on a major drive to win more basic civil rights and workplace protections in Southern and Western states where the rapid progress of the movement has largely eluded millions of gay men and lesbians.

The effort will shift tens of millions of dollars in the next few years to what advocates described as the final frontier for gay rights: states like Mississippi, Georgia, Arkansas and Texas, where Republicans dominate elected office and traditional cultural views on homosexuality still prevail.